糖果派对

Speaker: Edmund Heng (Institut des Hautes 脡tudes Scientifiques)


Abstract:

Classically, finite symmetries are captured by the action of a finite group. Moving to the quantum world, one has to allow for (possibly non-invertible) quantum symmetries 鈥� these are instead captured by the action of a more general algebraic structure, known as a fusion category. Such quantum symmetries are actually ubiquitous in mathematics; for example, given a category with an action of a finite group G (e.g. rep(Q), Coh(X) etc.), its G-equivariant category has instead the action of the category of representations rep(G), where rep(G) has the structure of a fusion category (and is not just a group when G is non-abelian).
The aim of this talk is to introduce fusion categories and discuss their role as 鈥渜uantum symmetries鈥� in relation to Bridgeland鈥檚 stability conditions. We first introduce a generalised notion replacing 鈥淕-invariant stability conditions鈥� in the setting of a triangulated category equipped an action of a fusion category C, which we will 鈥淐-equivariant stability conditions鈥�. The first result is that these stability conditions form a closed submanifold of the stability manifold, just as the G-invariant stability conditions do. Moreover, given a triangulated D with a G-action, so that its G-equivariant category D^G has a rep(G)-action, we will see the following (Morita) duality result for stability conditions: the complex manifold of G-invariant stability conditions (associated to D) is homeomorphic to the complex manifold of rep(G)-equivariant stability conditions (associated to D^G).

If time allows, I will discuss other more 鈥渆xotic鈥� actions of fusion categories on triangulated categories, and possibly its relation to Coxeter theory.

This is part of joint work with Hannah Dell and Anthony Licata.